In this part, we study equidistribution of the averaged measure \(\frac{1}{n}\sum _{1\leqslant k\leqslant n}\mu ^{*k}*\delta _x\)  for \(x\)  inside the bundle \(X_\mathcal {C}\) .
In fact, as we start by briefly explaining in §REF  below, all of them except the diagonal fibre action (Case 2.2) boils down to the corresponding results of Benoist–Quint [1]}, [2]}. The part §REF  is devoted to the diagonal fibre action case.
